Mathematics

Abstract A quick introduction to the techniques of derived categories and functors among them, with a focus on examples rather than proofs. Applications to classical algebraic geometry, that is schemes of finite type over an algebraically closed field.   Topics Categories of complexes, quasi isomorphisms, derived categories. Derived functors for half exact functors with enough acyclics. Naive cotangent complex, full cotangent complex. Left derived functors, Tor sheaves and groups, derived pullback. Ext groups for coherent sheaves on a projective scheme as left derived functors. Flatness for coherent sheaves wrt to projective morphisms in terms of derived categories. Alternative characterisations of étaleness and smoothness. Cohomology and base change. Dualising complex, Serre duality, Grothendieck–Serre duality.